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Let (X, d) be a metric space and let I be an interval in the real lineR. A function f : I → X is absolutely continuous on I if for every positive number ε, no matter how small, there is a positive number δ small enough so that whenever a sequence of pairwise disjoint sub-intervals [xk, yk] of I, k = 1, 2, ..., n satisfies

then

The collection of all absolutely continuous functions from I into X is denoted AC(I; X).

A further generalisation is the space ACp(I; X) of curves f : I → X such that

If μ and ν are measures on the same measure space (or, more precisely, on the same sigma-algebra) then μ is absolutely continuous with respect to ν if μ(A) = 0 for every set A for which ν(A) = 0. It is written as "μ << ν". In symbols:

If μ is a signed or complex measure, it is said that μ is absolutely continuous with respect to ν if its variation |μ| satisfies |μ| << ν; equivalently, if every set A for which ν(A) = 0 is μ-null.

The Radon-Nikodym theorem states that if μ is absolutely continuous with respect to ν, and ν is σ-finite, then μ has a density, or "Radon-Nikodym derivative", with respect to ν, which implies that there exists a ν-measurable function f taking values in [0,∞], denoted by f = dμ/dν, such that for any ν-measurable set A we have

The connection between absolute continuity of real functions and absolute continuity of measuresEdit

is locally an absolutely continuous real function. In other words, a function is locally absolutely continuous if and only if its distributional derivative is a measure that is absolutely continuous with respect to the Lebesgue measure.

has the Dirac delta distribution as its distributional derivative. This is a measure on the real line, a "point mass" at 0. However, the Dirac measure is not absolutely continuous with respect to Lebesgue measure , nor is absolutely continuous with respect to : but ; if is any open set not containing 0, then but .